Using Direct Proof. $1+2+3+\ldots+n = \frac{n(n + 1)}{2}$ I need help proving this statement. Any help would be great!
 A: As an alternative to induction you can try this:
Let $S_n = 1 + 2 + \ldots + n$.
Then $2S_n = \color{red}{1 + 2 + \ldots + n} + \color{blue}{1 + 2 + \ldots + n} = (\color{red}{1} + \color{blue}{n}) + (\color{red}{2} + \color{blue}{n-1}) + \ldots + (\color{red}{n} + \color{blue}{1}) = n(n+1)$
Therefore $$S_n = \frac{n(n+1)}{2}$$
A: Hint.
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          \b&\b&\b&\b&\b&\c&\c\cr\b&\b&\b&\b&\b&\b&\c\cr}$$
A: Here is an approach.
$$ s_n =1+2+3+\dots+(n-1)+n \\
   s_n =n+(n-1)+(n-2)+\dots+1 .  $$
Adding the above gives
$$2s_n = (1+n)+(2+(n-1))+(3+(n-2))+\dots+(1+n) $$
$$ =(1+n)+(1+n)+\dots+(1+n) $$
The above is nothing but adding $(1+n)$ n times and the result follows
$$ \implies s_n = \frac{n(n+1)}{2}. $$
A: Suppose you have a round robin tournament with $n+1$ players $a_1,\ldots,a_{n+1}$ who play one another exactly once. Let $S$ be the number of matches.
Each of the ${n+1}$ players plays the other $n$ players but you don't want to count duplicates (i.e. $a_1$ plays $a_2$ is the same match as $a_2$ plays $a_1$) so $S=\frac{1}{2}(n+1)n$.
On the other hand, you can enumerate the matches as follows:


*

*$n$ matches involving $a_1$,

*$n-1$ matches involving $a_2$ without involving $a_1$,

*$n-2$ matches involving $a_3$ without involving $a_2,a_1$,

*$\cdots$

*1 match involving $a_n$ without involving $a_{n-1},\ldots,a_1$.


So $S=n+(n-1)+\cdots+1$.
A: Below is a direct, inductive proof:
For $n=1$ we have 
$$1 = \frac{1(1+1)}{2}.$$
To proceed from $n=k$ to $n = k+1$ note that
$$1 + 2 + \cdots + k + (k+1) = \frac{k(k+1)}{2} + (k+1) = \frac{(k+1)(k+2)}{2},$$
qed.
